Is the location of the supremum of a stationary process nearly uniformly distributed?

It is, perhaps, surprising that the location of the unique supremum of a stationary process on an interval can fail to be uniformly distributed over that interval. We describe all possible distributions of the supremum location for broad class of such stationary processes. We show that, in the strongly mixing case, this distribution does tend to the uniform in certain sense as the length of the interval increases to infinity. We will see also that many of our results apply to other functionals of the sample paths, other than the location of the supremum, such as the first crossing times, and others.